some final work

thesis/chapters/appendix.tex

@@ -66,9 +66,9 @@ code is converted to an image using
 It has been mentioned several times that the implementation with
 \lstinline{ufuncs} as gates is faster than a pure \lstinline{python}
 implementation.  To support this statement a simple benchmark is written. The
-relatively simple Pauli $X$ gate us implemented, more complicated gates like $CX$ or $H$
+relatively simple Pauli $X$ gate is implemented, more complicated gates like $CX$ or $H$
 have worse performance when written in \lstinline{python}. The performance
-improvement when in this example is a factor around $6.4$.
+improvement in this example is a factor around $6.4$.
 One must note that the tested \lstinline{python} code is not
 realistic and in a possible application there would be a significant overhead.
 

thesis/chapters/conclusion.tex

@@ -3,9 +3,9 @@
 
 As seen in \ref{ref:performance} simulation using stabilizers is exponentially
 faster than simulating with dense state vectors. The graphical representation
-for stabilizer states is on average more efficiently than the stabilizer
-tableaux. In particular one can simulate more qbits while only applying
-Clifford gates.
+for stabilizer states is in realistic cases more efficiently than the
+stabilizer tableaux \cite{andersbriegel2005}. In particular one can simulate
+more qbits while only applying Clifford gates.
 
 This is considerably useful when working on quantum error correcting strategies
 as they often include many qbits; the smallest quantum error correcting
@@ -119,4 +119,5 @@ The exponential speedup of quantum computing is often attributed to
 entanglement, superposition and interference effects
 \cite{uwaterloo}\cite{21732}\cite{vandennest2018}. Stabilizer states show
 entanglement, superposition and interference effects; as do computations using
-general normalizers \cite{vandennest2018}.
+general normalizers \cite{vandennest2018}. The question why quantum computing
+can speed up computations exponentially is non-trivial.

thesis/chapters/implementation.tex

@@ -172,7 +172,13 @@ For the graphical state $(V, E, O)$ the list of vertices $V$ can be stored impli
 by demanding $V = \{0, ..., n - 1\}$. This leaves two components that have to be stored:
 The edges $E$ and the vertex operators $O$. Storing the vertex operators is
 done using a \lstinline{uint8_t} array. Every local Clifford operator is
-associated with an integer ranging from $0$ to $23$. Their order is
+associated with an integer ranging from $0$ to $23$
+\footnote{
+    \cite{andersbriegel2005} also uses integers from $0$ to $23$ to represent
+    the Clifford operators. The authors do not describe the mapping from integer
+    to operator and the order might be different.
+}
+. Their order is
 
 \begin{equation}
 \begin{aligned}
@@ -224,7 +230,8 @@ a vertex.  Also one must take care to keep the memory required to store a state
 low.  To meet both requirements a linked list-array hybrid is used.  For every
 vertex the neighbourhood is stored in a sorted linked list (which is a sparse
 representation of a column vector) and these adjacency lists are stored in
-a length $n$ array.
+a length $n$ array. \cite{andersbriegel2005} uses the same representation for
+the graph.
 
 Using this storage method all operations - including searching and toggling edges -
 inherit their time complexity from the sorted linked list.
@@ -240,7 +247,7 @@ three classes of operations.
 \begin{description} 
     \item[\hspace{-1em}]{\lstinline{RawGraphState.apply_C_L}\\
         This method implements local Clifford gates. It takes the qbit index
-        and the index of the local Clifford operator (ranging form $0$ to $23$).}
+        and the index of the local Clifford operator (ranging from $0$ to $23$).}
     \item[\hspace{-1em}]{\lstinline{RawGraphState.apply_CZ}\\
             Applies the $CZ$ gate to the state. The first argument is the
             act-qbit, the second the control qbit (note that this is just for
@@ -270,7 +277,7 @@ of the user.
 This implementation reads byte code from a file and executes it. The execution is 
 always done in three steps:
 
-\begin{enumerate}[1]
+\begin{enumerate}[1.]
     \item{Initializing the state according the header of the byte code file.}
     \item{Applying operations given by the byte code to the state. This includes local
             Clifford gates, $CZ$ gates and measurements (the measurement outcome is ignored).}
@@ -363,7 +370,7 @@ circuits are used. Length $m$ circuits are generated from the probability space
 with the uniform distribution. The continuous part $[0, 2\pi)$ is ignored when
 generating random circuits for the graphical simulator; in order to generate
 random circuits for dense vector simulations this is used as the argument $\phi$ of the
-$R_\phi$ gate.  For $m=1$ an outcome is mapped to a gate where 
+$R_\phi$ gate.  For $m=1$ an outcome is mapped to a gate with 
 
 \begin{equation}
 \begin{aligned}
@@ -382,7 +389,12 @@ dense vector simulator $S$ can be replaced by $R_\phi$ with the parameter $x$.
 
 Using this method circuits are generated and applied both to graphical and
 dense vector states and the time required to execute the operations
-\cite{timeit} is measured. The resulting graph can be seen in
+\cite{timeit} is measured\footnote{
+    The used computer had an \lstinline{Intel(R) Core(TM) i5-4590 CPU @ 3.30GHz} processor, 
+    \lstinline{7.7 GiB} RAM, and \lstinline{27 GiB} SSD swap space (which was unused).
+    The operating system was \lstinline{Linux 4.19.0-8-amd64 #1 SMP Debian 4.19.98-1 (2020-01-26) x86_64 GNU/Linux}.
+
+}. The resulting graph can be seen in
 Figure \ref{fig:scaling_qbits_linear} and Figure \ref{fig:scaling_qbits_log}.
 Note that in both cases the length of the circuits have been scaled linearly
 with the amount of qbits and the measured time was divided by the number of
@@ -421,11 +433,11 @@ the stabilizer tableaux like CHP \cite{CHP} with an average runtime behaviour
 of $\mathcal{O}\left(n\log(n)\right)$ instead of $\mathcal{O}\left(n^2\right)$.
 
 One should be aware that the gate execution time (the time required to apply
-a gate to the state) highly depends on the  state it is applied to. For the
+a gate to the state) highly depends on the  state it is applied to \cite{andersbriegel2005}. For the
 dense vector simulator and CHP this is not true: Gate execution time is
 constant for all gates and states. Because the graphical simulator has to
-toggle neighbourhoods the gate execution time of the $CZ$ gate varies
-significantly. The plot Figure \ref{fig:scaling_circuits_linear}
+toggle neighbourhoods the gate execution time of the $CZ$ and $M$ gates vary
+significantly \cite{andersbriegel2005}. The plot Figure \ref{fig:scaling_circuits_linear}
 shows the circuit execution time for two different numbers of qbits. One can observe three
 regimes:
 
@@ -455,7 +467,7 @@ regimes:
 
 These three regimes can be explained when considering the graphical states that
 typical live in these regimes. With increased circuit length the amount of
-edges increases which makes toggling neighbourhoods harder. Graphs from the
+edges increases which makes toggling neighbourhoods harder \cite{andersbriegel2005}. Graphs from the
 low-linear, intermediate and high-linear regime can be seen in Figure
 \ref{fig:graph_low_linear_regime}, Figure \ref{fig:graph_intermediate_regime}
 and Figure \ref{fig:graph_high_linear_regime}. Due to the great amount of edges

thesis/chapters/stabilizer.tex

@@ -16,6 +16,8 @@ This performance has since been improved to $n\log(n)$ time on average
 
 \section{Stabilizers and Stabilizer States}
 
+The discussion below follows the argumentation given in \cite{nielsen_chuang_2010}.
+
 \subsection{Local Pauli Group and Multilocal Pauli Group}
 
 \begin{definition}
@@ -46,7 +48,6 @@ deduced from its definition via the tensor product.
 
 \subsection{Stabilizers}
 
-The discussion below follows the argumentation given in \cite{nielsen_chuang_2010}.
 
 \begin{definition}
     \label{def:stabilizer}
@@ -292,6 +293,8 @@ and $s=0$ are obtained with probability $\frac{1}{2}$ and after choosing a $S^{(
 \section{The VOP-free Graph States}
 This section will discuss the vertex operator (VOP)-free graph states. Why they
 are called vertex operator-free will be clear in \ref{ref:sec_g_states}.
+Most of the discussion here is adapted from \cite{hein_eisert_briegel2008}
+whith some parts from \cite{andersbriegel2005}.
 
 \subsection{VOP-free Graph States}
 
@@ -650,7 +653,7 @@ studying its dynamics. The simplest case is a local Clifford operator $c_j$
 acting on a qbit $j$ changing the stabilizers to $\langle c_j S^{(i)}
 c_j^\dagger\rangle_i$. Using the definition of the graphical representation one
 sees that just the vertex operators are changed and the new vertex operators
-are given by
+are given by \cite{andersbriegel2005}
 
 \begin{equation}
     \begin{aligned}

thesis/main.tex

@@ -88,6 +88,20 @@ Supervised by Prof. Dr. Christoph Lehner}
 \bibliographystyle{unsrt} 
 \bibliography{main}{}
 
+\newpage
+{\Large\textbf{Acknowledgements}}
+\\
+\\
+I want to thank Prof. Dr. Christoph Lehner for offering me the possibility to
+write this thesis in his workgroup. He also found the time to answer my
+questions and help me when I got stuck while keeping an inspiring atmosphere.\\
+Further I own a big thank you to Simon Feyrer for proof-reading my thesis. \\
+Also I want to thank Andreas Hackl for proof-reading my thesis and helping out
+with several technical problems.\\
+Thanks to Shirley Galbaw for helping to fix some of my terrible ingris grammar.
+
+
+
 \newpage
 \textbf{Erklärung zur Anfertigung:}\\
 Ich habe die Arbeit selbständig verfasst, keine anderen als die angegebenen